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Owner: jcarruthers
Created: Nov 6, 2018
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A few years back, Idaho joined states that require students take a college entrance exam for graduation.  Furthermore, the state of Idaho sets aside a day each year for juniors to take the exam paid for by the state.  While students could take the ACT instead of the SAT, the SAT has become the exam taken by almost all juniors in the state (93% in 2017).  Since Idaho adopted this requirement, the state's average score has decreased significantly.  Because of this, I wanted to explor the association with average score and percent taking the test.

Linear Model:
This scatterplot shows a strong, negative association, but it does look to have a curve to it.

Result 1: 2017 SAT Regression Scatterplott   [Info]

The correlation, R = -0.8675, confirms a strong linear assoiation.  Whith R= 75.3% we can say that 75.3% of the variation in state SAT scores can be explained by percent taking the test.  The slope shows us that, on average, a states score drops 2.27 points for each percent increase in those taking the tesc.

Result 2: 2017 SAT Regression Results   [Info]
Simple linear regression results:
Dependent Variable: Total
Independent Variable: Participation
Total = 1216.6391 - 2.2746766 Participation
Sample size: 51
R (correlation coefficient) = -0.86753981
R-sq = 0.75262533
Estimate of error standard deviation: 46.470993

Parameter estimates:
ParameterEstimateStd. Err.AlternativeDFT-StatP-value
Intercept1216.63919.8657295 ≠ 049123.31973<0.0001
Slope-2.27467660.1862987 ≠ 049-12.209836<0.0001

Analysis of variance table for regression model:
SourceDFSSMSF-statP-value
Model1321946.4321946.4149.0801<0.0001
Error49105818.112159.5532
Total50427764.51

The residual plot has a definate curved pattern to it.  This sugests we might be able to find a better model.

Result 3: 2017 SAT Regression Residual Plot   [Info]

Power Model:
By re-expressing both x and y using the natural logarithm, the scatterplot shows a stronger linear model than before.

Result 4: 2017 SAT Regression Scatterplot ln(y) ln(x)   [Info]

The correlation, R = 0.9141, has also improved significantly.  As has R= 83.6%.

Result 5: 2017 SAT Regression Results ln(y) ln(x)   [Info]
Simple linear regression results (w/ transformation):
Dependent Variable: ln(Total)
Independent Variable: ln(Participation)
ln(Total) = 7.1785279 - 0.052376794 ln(Participation)
Sample size: 51
R (correlation coefficient) = -0.91407
R-sq = 0.83552397
Estimate of error standard deviation: 0.033396834

Parameter estimates:
ParameterEstimateStd. Err.AlternativeDFT-StatP-value
Intercept7.17852790.010897076 ≠ 049658.75728<0.0001
Slope-0.0523767940.0033198061 ≠ 049-15.777064<0.0001

Analysis of variance table for regression model:
SourceDFSSMSF-statP-value
Model10.277627820.27762782248.91575<0.0001
Error490.0546520780.0011153485
Total500.3322799

The residual plot shows no pattern but there might be one outlier.  Because this possible outlier is near the center of the data, it only has a little leverage.  Therefore it might make the correlation look weaker than it is, but it will have little affect on the slope.

Result 6: 2017 SAT Regression Residual ln(y) ln(x)   [Info]

Data set 1. 2017 SAT Scores by State   [Info]